What mathematical skill is the textbook trying to teach with this image?

Pseudocontext Saturday #11

Calculating angles of rotation (57%, 249 Votes)

Graphing rational functions (28%, 121 Votes)

Proving trigonometric identities (15%, 67 Votes)

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Every Saturday, I post an image from a math textbook. It’s an image that implicitly or explicitly claims that “this is how we use math in the world!”

I post the image without its mathematical connection and offer three possibilities for that connection. One of them is the textbook’s. Two of them are decoys. You guess which connection is real.

After 24 hours, I update the post with the answer. If a plurality of the commenters picks the textbook’s connection, one point goes to Team Commenters. If a plurality picks one of my decoys, one point goes to Team Me. If you submit a mathematical question in the comments about the image that isn’t pseudocontext, collect a personal point.

Later the text goes on to ask students to construct and graph the formula for the average cost of producing the Mega-Grandma, which turns out to be a rational function given the constraints.

But the text might have been better served by asking students to solve for generic widgets, or tennis balls, or something a little less gonzo-bananas than the Mega-Grandma exoskeleton, which is all I’m going to remember from this unit in the textbook.

7 Comments

Alex Gladney-Lemon

I’m torn on this entire pseudo-context thread. On one hand, I agree that it can be “fun” to point out the absurdity of certain math tasks. On the other, actual people wrote these tasks, and it seems mean to turn a critique of their work into a literal (and weekly) game.

That said, I also find the exercise somewhat ironic. As I understand it, one of your motivations is to *caution* teachers against treating the real-world as a panacea. You write:

I hope this series will serve to remind us weekly of the madness that lies at the extreme end of a position that says ‘students will only be interested in mathematics if it’s real world. The end of that position leads to dog bandanas and other bizarre connections which serve to make math seem less real to students and more alien, a discipline practiced by weirdos and oddballs.

That’s a good point, and one worth stressing. And so on that note, let me ask you: Have you ever seen a square swimming pool?

To clarify, the question about the swimming pool is a genuine one, and gets to a larger question I continue to have about what makes a context “pseudo.” In an earlier post, you defined a pseudo-contextual task as one in which:

1. The task asks a different question than the one most people would naturally ask, and/or
2. The task prompts students to use math that they wouldn’t otherwise use.

Using these criteria, the pool task is certainly more authentic than is, say, the robot example. How many tiles? is the question I’d ask, and an algebraic expression is the tool I’d use to answer it. Meanwhile, there isn’t anything in the robot description that screams “rational functions” to me.

That said, these criteria don’t seem sufficient to me, or rather, they seem to jump the gun. When we label the robot task as a “pseudo-context,” what we’re really critiquing is how the authors use the context. But Mega-Grandma? It’s a thing. Minimizing manufacturing costs? That’s a problem. The developers may have asked bad questions, but the situation itself is legit.

Contrast this with the pool problem. If the robot problem is an example of “bad questions about a real situation,” the pool problem seems to me like an example of “good questions about an invented situation.” I say that for two reasons. First, I can’t recall ever having seen a square swimming pool. Second, even if what we’re talking about is a hot tub, I have a hard time imagining that technicians use (4n – 4) to calculate how many tiles to buy. Instead, I imagine that they buy tiles in bulk and, if the tiles don’t fit the perimeter perfectly, that they use grout.

I don’t think this means the the pool task is bad. On the contrary, I think it’s a wonderful activity for helping students understand how to write algebraic expressions. Still, given that “pseudo” literally means “not genuine,” I have a hard time seeing any fundamental difference in authenticity between the robot task and the pool one. They both involve invention, just at different steps in the process: one given the context, and the other to get the context.

Which brings me back to my question about the two criteria above. My own preference for authentic applications notwithstanding, I think you make a valuable point about the dangers of treating “real world” as a magic bullet, and I agree that many math tasks are so forced as to be laughable. At the same time, I notice that most of the tasks you’ve developed recently are grounded in some ostensibly real-world situation, whether a taco cart or pennies or a pool. If we take a context as given, your two criteria make sense to me. Yet before we ask whether a certain context prompts students to naturally answer the natural question, is it important to first ask whether the context itself is a genuine one? Before 1 and 2 above, should there be some Criteria 0: If the situation is grounded in a real-world context, does that context actually exist in the form it’s presented?

Dan Meyer

… it seems mean to turn a critique of their work into a literal (and weekly) game.

Well there’s no way I’m disciplined enough to do anything on a weekly basis, so no one should worry too much.

Before 1 and 2 above, should there be some Criteria 0: If the situation is grounded in a real-world context, does that context actually exist in the form it’s presented?

Maybe. My guess is that if I were to actually locate a square swimming pool, it wouldn’t change how many students interacted with the task.

Another criteria that interests me is to ask: “Is there interesting and non-routine work to do in this context, or is the author banking on the context to provide all of the interest?” If we stripped away the context, is there anything of interest here? My guess is that we dispensed with the “pool” context entirely and just said, “hey check out this pattern,” we’d lose some students but not many. There is still interesting and non-routine work to do.

I’ve been trying to come up with a clean way to distinguish between conceptual understanding tasks and applications for a while. It’s proven trickier than I expected — more of a gradient, perhaps, than a line — but the question of where a task comes from keeps popping up: namely, was the motivation to use math to explain something in the external world, or was is to illustrate some underlying mathematical concept (and a “real world” context simply provided helpful grounding). Somewhere in there, I suspect my question about the pool is related to this.

I’m interested in your observation that if you ditched the pool context, most kids would still dig the lesson. I expect you’re right; it reminds me of the snake pattern video from Sadie’s classroom. In that case, if what you’re after is the “puzzling,” and if the animated pattern sparks that already, why use the pool context at all? Perhaps the tiles lingo makes it easier for kids to talk about, but are you concerned at all that the pool frame (1) risks clouding the appreciation of math for its own sake, and/or (2) risks setting kids up to say, “But this isn’t how pools are actually designed?”

I’d love to know more about how you balance those two ideas — concreteness and abstraction — and how you decide whether a context will facilitate the process vs. get in the way.

Totally agree, BTW, with your added criteria. I’m actually re-going through our library now, and am seeing some examples where we leaned way too hard on the context. An excellent, excellent point.

Dan Meyer

In that case, if what you’re after is the “puzzling,” and if the animated pattern sparks that already, why use the pool context at all?”

The pool and its border are useful mental tools. They give us a common language (the lingo you mentioned) and concept so we can do interesting work. Without the pool, I’d have to add extra clarification about where the squares go. With the pool, I don’t have to, because students know we aren’t going to put those tiles in the water. It’s a helpful analogy.

Are you concerned at all that the pool frame (1) risks clouding the appreciation of math for its own sake, and/or (2) risks setting kids up to say, “But this isn’t how pools are actually designed?”

Maybe I should be. I haven’t heard that objection from students or their teachers, though. I think that’s because, looking up to my earlier comment, the lesson doesn’t present itself as being “a job connection: how contractors use math in their jobs.” I’ll keep my ears open, though.